Converting Units

Study Sheet

Converting Units

Metric prefixes and time

Multiply or Divide?

Concept
The Big Idea

Going to a smaller unit means you need more of them, so multiply. Going to a bigger unit means fewer, so divide.

Converting units is multiplying by 11 in disguise: since 11 foot =12= 12 inches, the fraction 12 in1 ft\tfrac{12\text{ in}}{1\text{ ft}} equals 11. Multiply 33 feet by it and the feet cancel, leaving 3636 inches. Write the conversion so the unit you want to remove is on the opposite side of the fraction.

Example
Metres to centimetres
00110022003300mcm
Each metre lines up with 100 more centimetres.

One metre is 100100 centimetres, so 33 metres is 3×100=3003 \times 100 = 300 centimetres — more of a smaller unit.

Example
Grams to kilograms

11 kg =1000= 1000 g. Going to a bigger unit means fewer of them, so divide: 3500÷1000=3.53500 \div 1000 = 3.5. So 35003500 g =3.5= 3.5 kg.

Side note
Where it goes wrong

Multiplying when you should divide (or the reverse). Let the units decide: if the units do not cancel, the fraction is upside down.

Try it
Try it: feet and inches

Convert 77 feet to inches, and 4848 inches to feet. (11 ft =12= 12 in.)

Show the answer

Feet to inches is big to small, so multiply: 7×12=847 \times 12 = \mathbf{84} in. Inches to feet is small to big, so divide: 48÷12=448 \div 12 = \mathbf{4} ft.

In Plain Terms & More Examples

Concept
In plain terms

Changing to a smaller unit needs more of them, so multiply; changing to a bigger unit needs fewer, so divide.

Example
Kilometres to metres

One kilometre is 10001000 metres, so 55 km =5×1000=5000= 5 \times 1000 = 5000 m.

Example
Minutes to hours
00601120218032404minh
240 minutes lines up with 4 hours.

One hour is 6060 minutes, so 240240 min =240÷60=4= 240 \div 60 = 4 hours.

Side note
Where it goes wrong

Moving the point the wrong way. 55 km is not 0.0050.005 m: a metre is smaller than a kilometre, so there must be more metres, 50005000. Ask "should the number get bigger or smaller?" before calculating.

Try it
Try it: litres and hours

Convert 2.42.4 L to millilitres and 9090 minutes to hours.

Show the answer

11 L =1000= 1000 mL, so 2.4×1000=24002.4 \times 1000 = \mathbf{2400} mL. 11 h =60= 60 min, so 90÷60=1.590 \div 60 = \mathbf{1.5} h, an hour and a half.

Two-Step Conversions

Concept
Chain the steps

To go from hours to seconds, go through minutes: 11 h =60= 60 min =60×60=3600= 60\times60 = 3600 s. Each step multiplies (bigger unit to smaller) or divides (smaller to bigger).

Example
Kilometers to centimeters
kmmcm× 1000× 100
Each step to a smaller unit multiplies.

2.52.5 km =2500= 2500 m =250,000= 250{,}000 cm.

Example
Hours to seconds

22 h =2×60=120= 2 \times 60 = 120 min, and 120120 min =120×60=7200= 120 \times 60 = 7200 s. Two steps, each multiplying by 6060.

Side note
Where it goes wrong

Treating time like a decimal. 1.51.5 hours is not 11 hour 5050 minutes: the .5.5 is half an hour, 3030 minutes, so 1.51.5 h =90= 90 min. Minutes count up to 6060, not 100100.

Try it
Try it: convert

Convert 33 hours 2020 minutes to minutes, and 45004500 g to kilograms.

Show the answer

3×60+20=2003\times60 + 20 = \mathbf{200} minutes. 4500÷1000=4.54500 \div 1000 = \mathbf{4.5} kg.

Word Problems

Concept
Convert first, then compute

Before adding, subtracting, dividing or comparing, put every measurement into the same unit. Then the arithmetic is ordinary.

22 m and 3535 cm cannot be combined as "2+352 + 35", just as 22 dollars and 3535 cents are not 3737 of anything. Once both are in centimetres, 200+35=235200 + 35 = 235 cm makes sense.

Example
Juice boxes

Each juice box holds 250250 mL. Six boxes hold 15001500 mL =1.5= 1.5 L.

Example
Cutting a board
25252525252525252 m = 200 cm
Eight 25 cm pieces fill 200 cm.

A 22 m board is cut into 2525 cm pieces. How many pieces? Change to one unit: 22 m =200= 200 cm. Then 200÷25=8200 \div 25 = 8 pieces.

Side note
Where it goes wrong

Leaving mixed units un-regrouped. 11 m 2020 cm +90+ 90 cm is not "11 m 110110 cm" as a final answer: 110110 cm is 11 m 1010 cm, so the total is 22 m 1010 cm.

Try it
Try it: the race

A race is 55 km. Lena has run 32503250 m. How many kilometers are left?

Show the answer

5000−3250=17505000 - 3250 = 1750 m =1.75= \mathbf{1.75} km.

Customary Units

Concept
Not powers of ten

In U.S. customary units: 1212 in =1= 1 ft, 33 ft =1= 1 yd, 1616 oz =1= 1 lb, and 44 qt =1= 1 gal. The rule is the same as in metric (bigger unit to smaller: multiply), but the numbers are not powers of ten.

Because 1212, 33, 1616 and 44 are not tens, you cannot just move a decimal point. Write the conversion as "11 ft =12= 12 in" and ask how many of the new unit fit into one of the old.

Example
Yards to inches
00132639412515ydft
Every yard is 3 more feet.

55 yd =5×3=15= 5 \times 3 = 15 ft, and 1515 ft =15×12=180= 15 \times 12 = 180 in.

Example
Pounds and ounces

33 lb 44 oz in ounces: 3×16=483 \times 16 = 48 oz, plus the 44 oz, is 5252 oz.

Example
Quarts to gallons

44 qt make a gallon, and 10=4×2+210 = 4 \times 2 + 2, so 1010 qt =2= 2 gal 22 qt. Since 22 qt is half a gallon, that is also 2.52.5 gal.

Side note
Where it goes wrong

Using decimals as if the units were tens. 2.52.5 ft is not 22 ft 55 in: 0.50.5 ft is half of 1212 in, which is 66 in. So 2.52.5 ft =2= 2 ft 66 in =30= 30 in.

Try it
Try it: pounds and yards

How many ounces are in 2122\tfrac12 lb? How many feet are in 44 yd?

Show the answer

212×16=32+8=402\tfrac12 \times 16 = 32 + 8 = \mathbf{40} oz. 4×3=124 \times 3 = \mathbf{12} ft.

Why It Works

Because 1212 inches and 11 foot are the same length, the ratio 12 in1 ft\tfrac{12\text{ in}}{1\text{ ft}} is exactly 11, and multiplying any quantity by 11 leaves it unchanged — only its name changes. Chaining several such fractions converts through intermediate units (km→m→cm\text{km} \to \text{m} \to \text{cm}) without ever deciding whether to multiply or divide: the cancellation of units does the deciding.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.